A local-global principle for isogenies of prime degree over number fields

We give a description of the set of exceptional pairs for a number field $K$, that is the set of pairs $(\ell, j(E))$, where $\ell$ is a prime and $j(E)$ is the $j$-invariant of an elliptic curve $E$ over $K$ which admits an $\ell$-isogeny locally almost everywhere but not globally. We obtain an upper bound for $\ell$ in such pairs in terms of the degree and the discriminant of $K$. Moreover, we prove finiteness results about the number of exceptional pairs.Comment: 22 pages, presentation improved as suggested by the referees. To appear in Journal of London Mathematical Society. arXiv admin note: text overlap with arXiv:1006.1782 by other author

On Serre's uniformity conjecture for semistable elliptic curves over totally real fields

Let $K$ be a totally real field, and let $S$ be a finite set of non-archimedean places of $K$. It follows from the work of Merel, Momose and David that there is a constant $B_{K,S}$ so that if $E$ is an elliptic curve defined over $K$, semistable outside $S$, then for all $p>B_{K,S}$, the representation $\bar{\rho}_{E,p}$ is irreducible. We combine this with modularity and level lowering to show the existence of an effectively computable constant $C_{K,S}$, and an effectively computable set of elliptic curves over $K$ with CM $E_1,\dotsc,E_n$ such that the following holds. If $E$ is an elliptic curve over $K$ semistable outside $S$, and $p>C_{K,S}$ is prime, then either $\bar{\rho}_{E,p}$ is surjective, or $\bar{\rho}_{E,p} \sim \bar{\rho}_{E_i,p}$ for some $i=1,\dots,n$.Comment: 7 pages. Improved version incorporating referee's comment

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+y2m=zpx^{2\ell}+y^{2m}=z^p

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular. Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To a putative non-trivial primitive solution of the generalized Fermat $x^{2\ell}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}(\zeta_p)^+$, and study its mod $\ell$ representation with the help of level lowering and our modularity result. We deduce the non-existence of non-trivial primitive solutions if $p \le 11$, or if $p=13$ and $\ell$, $m \ne 7$.Comment: Introduction rewritten to emphasise the new modularity theorem. Paper revised in the light of referees' comment

Constructing hyperelliptic curves with surjective Galois representations

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod l Galois representations. The main result of the paper is the following. Suppose n=2g+2 is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than n (this hypothesis appears to hold for all g different from 0,1,2,3,4,5,7 and 13). Then there is an explicit integer N and an explicit monic polynomial $f_0(x)\in \mathbb{Z}[x]$ of degree n, such that the Jacobian $J$ of every curve of the form $y^2=f(x)$ has $Gal(\mathbb{Q}(J[l])/\mathbb{Q})\cong GSp_{2g}(\mathbb{F}_l)$ for all odd primes l and $Gal(\mathbb{Q}(J[2])/\mathbb{Q})\cong S_{2g+2}$, whenever $f(x)\in\mathbb{Z}[x]$ is monic with $f(x)\equiv f_0(x) \bmod{N}$ and with no roots of multiplicity greater than $2$ in $\overline{\mathbb{F}}_p$ for any p not dividing N.Comment: 24 pages, minor correction

Residual Representations of Semistable Principally Polarized Abelian Varieties

Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let $\bar{\rho}_{A,\ell} : G_{\mathbb{Q}} \rightarrow \mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the representation giving the action of $G_{\mathrm{Q}} :=\mathrm{Gal}(\bar{\mathrm{Q}}/\mathrm{Q})$ on the $\ell$-torsion group $A[\ell]$. We show that if $\ell \ge \max(5,d+2)$, and if image of $\bar{\rho}_{A,\ell}$ contains a transvection then $\bar{\rho}_{A,\ell}$ is either reducible or surjective. With the help of this we study surjectivity of $\bar{\rho}_{A,\ell}$ for semistable principally polarized abelian threefolds, and give an example of a genus $3$ hyperelliptic curve $C/\mathbb{Q}$ such that $\bar{\rho}_{J,\ell}$ is surjective for all primes $\ell \ge 3$, where $J$ is the Jacobian of $C$.Comment: Paper has appeared in Research in Number Theor

Deep congruences + the Brauer-Nesbitt theorem

We prove that mod-$p$ congruences between polynomials in $\mathbb Z_p[X]$ are equivalent to deeper mod-$p^{1+v_p(n)}$ congruences between the $n^{\rm th}$ power-sum functions of their roots. We give two proofs, one combinatorial and one algebraic. This result generalizes to torsion-free $\mathbb Z_{(p)}$-algebras modulo divided-power ideals. As a direct consequence, we obtain a refinement of the Brauer-Nesbitt theorem for finite free $\mathbb Z_p$-modules with an action of a single linear operator, with applications to the study of Hecke modules of mod-$p$ modular forms

Images of Galois representations

Dans cette thèse, on étudie les représentations 2-dimensionnelles continues du groupe de Galois absolu d'une clôture algébrique fixée de Q sur les corps finis qui sont modulaires et leurs images. Ce manuscrit se compose de deux parties.Dans la première partie, on étudie un problème local-global pour les courbes elliptiques sur les corps de nombres. Soit E une courbe elliptique sur un corps de nombres K, et soit l un nombre premier. Si E admet une l-isogénie localement sur un ensemble de nombres premiers de densité 1 alors est-ce que E admet une l-isogénie sur K ? L'étude de la repréesentation galoisienne associéee à la l-torsion de E est l'ingrédient essentiel utilisé pour résoudre ce problème. On caractérise complètement les cas où le principe local-global n'est pas vérifié, et on obtient une borne supérieure pour les valeurs possibles de l pour lesquelles ce cas peut se produire.La deuxième partie a un but algorithmique : donner un algorithme pour calculer les images des représentations galoisiennes 2-dimensionnelles sur les corps finis attachées aux formes modulaires. L'un des résultats principaux est que l'algorithme n'utilise que des opérateurs de Hecke jusqu'à la borne de Sturm au niveau donné n dans presque tous les cas. En outre, presque tous les calculs sont effectués en caractéristique positive. On étudie la description locale de la représentation aux nombres premiers divisant le niveau et la caractéristique. En particulier, on obtient une caractérisation précise des formes propres dans l'espace des formes anciennes en caractéristique positive.On étudie aussi le conducteur de la tordue d'une représentation par un caractère et les coefficients de la forme de niveau et poids minimaux associée. L'algorithme est conçu à partir des résultats de Dickson, Khare-Wintenberger et Faber sur la classification, à conjugaison près, des sous-groupes finis de PGL₂(F¯ℓ). On caractérise chaque cas en donnant une description et des algorithmes pour le vérifier. En particulier, on donne une nouvelle approche pour les représentations irréductibles avec image projective isomorphe soit au groupe symétrique sur 4 éléments ou au groupe alterné sur 4 ou 5 éléments.In this thesis we investigate 2-dimensional, continuous, odd, residual Galois representations and their images. This manuscript consists of two parts.In the first part of this thesis we analyse a local-global problem for elliptic curves over number fields. Let E be an elliptic curve over a number field K, and let ℓ be a prime number. If E admits an ℓ-isogeny locally at a set of primes with density one then does E admit an ℓ-isogeny over K? The study of the Galois representation associated to the ℓ-torsion subgroup of E is the crucial ingredient used to solve the problem. We characterize completely the cases where the local-global principle fails, obtaining an upper bound for the possible values of ℓ for which this can happen. In the second part of this thesis, we outline an algorithm for computing the image of a residual modular 2-dimensional semi-simple Galois representation. This algorithm determines the image as a finite subgroup of GL₂(F¯ℓ), up to conjugation, as well as certain local properties of the representation and tabulate the result in a database. In this part of the thesis we show that, in almost all cases, in order to compute the image of such a representation it is sufficient to know the images of the Hecke operators up to the Sturm bound at the given level n. In addition, almost all the computations are performed in positive characteristic.In order to obtain such an algorithm, we study the local description of the representation at primes dividing the level and the characteristic: this leads to a complete description of the eigenforms in the old-space. Moreover, we investigate the conductor of the twist of a representation by characters and the coefficients of the form of minimal level and weight associated to it in order to optimize the computation of the projective image.The algorithm is designed using results of Dickson, Khare-Wintenberger and Faber on the classification, up to conjugation, of the finite subgroups of PGL₂(F¯ℓ). We characterize each possible case giving a precise description and algorithms to deal with it. In particular, we give a new approach and a construction to deal with irreducible representations with projective image isomorphic to either the symmetric group on 4 elements or the alternating group on 4 or 5 elements

A note on the minimal level of realization for a mod ℓ eigenvalue system

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